One of the advantages of using oils for beauty care are that they are often readily available, possibly even sitting there in your kitchen cabinet right now. They don't need preservatives (though added antioxidants extend their shelf-life with no harm to you), and you only need a small amount at a time. They're best to use when you get out of the shower or after you wash your face, since they don't have water built into them like lotions and creams do. They're one of the easiest ways to simplify your beauty routine, and you don't have to read through long lists of ingredients, or figure out which of those many ingredients is the one bothering you. There are also many other beauty oils out there, but these are ones you might come by more readily.
When using oils, make sure to start with just a tiny amount, so you don't end up feeling greasy. I'm concentrating mostly on the face here, since your it's mostly likely to have problems that can take awhile to clear up- namely in the form of acne. If your skin is dry and not likely to breakout, you can experiment with different oils to see which ones work best for you. Transfer some oil into a small bottle, preferably one with a dispensing cap, so you can keep your cooking oils separate from you beauty oils. Jars would work better for coconut oil and solid butters. When buying oils, look for ones labeled "cold pressed."
Some popular grocery store oils that you can use on your hair and skin include olive, coconut, and avocado. I wouldn't use coconut oil on your face unless you're not at all prone to break outs. It does have a long shelf life, so it could to stock up if you like it for your hair and body. Olive oil seems to have different comedogenic ratings depending on where I looked, so you might want to skip that one as well. Avocado has a low-medium comedogenicity rating, so it could be suitable for most people. It does have a shorter shelf life though, so buy small amounts and keep what you're not using in the refrigerator.
If you can find them, jojoba oil and hemp seed oil are non-comedogenic. Jojoba has an indefinite shelf life, so you don't have to worry about it going rancid, but it is pricey. Trader Joe's sometimes has small bottles, so it might be worth picking up if you see it to see how you like it. Hemp seed oil has a short shelf life, so this is another one to keep the main bottle in the refrigerator.
If you see shea butter in the beauty section, it's also non-comedogenic and great for your skin and hair. Make sure it's just shea butter, though some added essential oils or vitamin E are fine. Look for unrefined shea to get the maximum benefits from it- it might have more of an odor and color to it, but it will be better for you. Cocoa butter has a high melting point, so it's not so easy to use, plus it's highly comedogenic, so unless you really want that chocolaty odor, you might want to look elsewhere.
If you suffer from oily skin, hazelnut oil is the one for you- it is slightly comedogenic, so patch test before smearing it all over your face. Putting oil on oily skin might sound silly, but it is slightly astringent, balancing your skin instead of stripping it of those excess oils. This one might be harder to find, but fancier grocery stores might stock it.